Apparently they do, I've occasionally posted about a problem I was having and had someone tell me they opened a ticket on it and then it's followed up by email.
But probably not at the moment. I think it's a national holiday in the US today so if anyone at ZOS is working it's likely only a skeleton crew dealing with the most urgent issues.
In general I'd say if you have a problem you need support to fix it's better to contact them directly (via the link at the top of the page) than to post on the forum and hope the relevant person sees it. I only post issues on the forum if I want to know if it's affecting everyone else or if I think it's a problem on my end which I can fix but I'm not sure how.